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An analytical model for strength of jointed rock masses Hossein Rafiei Renania, , C. Derek Martinb, Ming Caic,d ⁎

a

Klohn Crippen Berger, Vancouver, Canada Department of Civil & Environmental Engineering, University of Alberta, Edmonton, Canada Geomechanics Research Centre, MIRARCO, Laurentian University, Sudbury, Canada d Bharti School of Engineering, Laurentian University, Sudbury, Canada b c

ARTICLE INFO

ABSTRACT

Keywords: Analytical failure criteria Analytical GSI chart Joint persistence Hoek-Brown Numerical analysis

An analytical model of rock mass with non-persistent joints is utilized to develop linear and nonlinear failure criteria based on intact rock strength, joint shear strength and joint persistence. The analytical model matches the solutions for the strength of fracture-free and fully jointed rocks and provides a reasonable transition of strength for rocks with non-persistent joints. Consistent with empirical evidence, the model predicts that increasing joint intensity and decreasing joint shear strength cause a reduction in the cohesive and frictional components of rock mass strength. Joint persistence in the analytical model has not only a physical definition as the normalized jointed area, but also a mechanical interpretation as the normalized reduction in strength due to the presence of joints. The relationship between the analytical model and the empirical Hoek-Brown failure criterion with Geological Strength Index (GSI) is explored. The analytical model of rock mass is utilized to develop and calibrate a quantitative GSI chart. The analytical model is validated using the results of comprehensive tests on undisturbed samples of naturally jointed Panguna andesite and thermally granulated Carrara marble. Practical application of the analytical model is illustrated by numerical back analysis of displacements around the Nathpa Jhakri powerhouse cavern.

1. Introduction Design of engineering structures requires adequate knowledge of the mechanical properties of engineering materials. The overwhelming amount of experimental and theoretical research in mechanical and civil engineering has made it possible to capture the behavior of manufactured materials such as steel and concrete with relatively high accuracy. The lack of natural discontinuities in steel and concrete structures allows the results of tests on laboratory scale samples to be directly used in engineering design. In rock engineering, the subject of interest is rock mass comprised of intact rock and embedded joints (Fig. 1). Although the micro-structure of intact rock is more heterogeneous than manufactured materials, the behavior of intact rock at laboratory scale has been extensively studied and can be accurately captured using standard methods of testing and analysis (e.g., Bieniawski, 1967, 1974; Hoek and Brown, 1980; Haimson and Chang, 2000; Mogi, 2007). Accurate knowledge of intact rock properties alone, however, is of limited value in rock engineering because rock mass behavior is largely governed by the geometric and mechanical properties of discontinuities. Determination of rock mass strength has been a primary objective in

⁎

rock mechanics and rock engineering since the establishment of the International Society for Rock Mechanics (ISRM) in 1962. Despite significant efforts, strength of rock masses at large scale has remained a less understood subject. It is partly because of the inherent structural complexity of rock masses with fractures of various scales and properties at different orientations. The combination of intact rock properties and fracture network characteristics are typically specific to a given rock mass which makes it difficult to develop general solutions. In addition, testing of representative undisturbed samples of rock mass at large scale is extremely challenging and reliable experimental data are very rare. The current practical approach for crude estimation of rock mass strength is based on limited empirical data from testing of large scale samples and back analysis of rock structures (e.g., Diederichs, 2007; Hoek and Brown, 1980; Rafiei Renani and Martin, 2018a,b; Hoek and Brown, 2019; Hoek et al., 2002; Rafiei Renani et al., 2019). Discontinuum numerical models of rock mass have been presented which to some extent capture the complex nature of rock mass behavior (e.g., Potyondy and Cundall, 2004; Cho et al., 2007; Mas Ivars et al., 2011; Potyondy, 2014). However, such models are challenging to develop and analyze, and the results are applicable only to the given rock mass used

Corresponding author. E-mail address: [email protected] (H. Rafiei Renani).

https://doi.org/10.1016/j.tust.2019.103159 Received 20 July 2019; Received in revised form 18 October 2019; Accepted 19 October 2019 0886-7798/ © 2019 Elsevier Ltd. All rights reserved.

Please cite this article as: Hossein Rafiei Renani, C. Derek Martin and Ming Cai, Tunnelling and Underground Space Technology, https://doi.org/10.1016/j.tust.2019.103159

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Nomenclature

i

j rm

n 1

3 1, i 1, j

1, rm

c

ci cj crm i

Friction angle of joint Friction angle of rock mass mi , si Hoek-Brown constants for intact rock Hoek-Brown constants for joint mrm , srm, arm Hoek-Brown constants for rock mass GSI Geological strength index D Disturbance factor p Area-based joint persistence pl Length-based joint persistence Mi Intact rock strength mobilization factor Mj Joint strength mobilization factor Ni Number of intact rock elements Nj Number of jointed rock elements Angle between the plane of joint and direction of 1 Angle between the plane of failure and direction of Instantaneous friction angle j

rm

Shear stress Shear strength of intact rock Shear strength of joint Shear strength of rock mass Normal effective stress Major effective principal stress Minor effective principal stress Strength of intact rock Strength of joint Strength of rock mass Uniaxial compressive strength of intact rock Cohesion of intact rock Cohesion of joint Cohesion of rock mass Friction angle of intact rock

for model calibration. In this study, a conceptual model of rock mass with non-persistent joints is presented and analytical solutions are developed for rock mass strength using linear and nonlinear failure criteria. The model can be used to investigate the influence of joint persistence and shear strength on rock mass strength. The analytical model is compared with the empirical Hoek-Brown failure criterion leading to development of a calibrated quantitative GSI chart. The analytical model is validated using experimental data and used in numerical analysis of a powerhouse cavern.

3

2.1. Mohr-Coulomb failure criterion The Mohr-Coulomb failure criterion can be considered as an extension of the Amonton’s law of friction which states that the shear force required for sliding on the contact surface between two bodies is proportional to the normal force. For bonded surfaces, an additional cohesive component can be added to the stress dependent frictional strength. Hence, shear strength of joints under effective normal stress n can be given by: j

= cj +

n

tan ( j )

(1)

where j , cj , and j are shear strength, cohesion and friction angle of the joint, respectively. Note that all stresses used throughout this paper are effective stresses unless stated otherwise. Application of the Mohr-Coulomb failure criterion can be extended to solid materials with the interpretation that failure occurs on a critically oriented plane where shear stress reaches a critical threshold governed by the cohesive and frictional strength components. Therefore, the shear strength of intact rock can be given by:

2. Failure criteria in rock engineering Over the years, numerous failure criteria have been developed by various researchers to determine the strength of intact rock, joints, and rock masses (e.g., Mogi, 1967, 1971; Jennings, 1970; Bieniawski, 1974; Lade and Duncan, 1975; Barton and Choubey, 1977; Sheorey et al., 1989; Zhou, 1994; Benz et al., 2008; Rafiai, 2011; Rafiai and Jafari, 2011; Rafiai et al., 2013). However, most of these failure criteria are seldom used in practical rock engineering and remain of academic interest only. The Mohr-Coulomb and Hoek-Brown criteria are by far the most commonly used failure criteria in rock engineering and therefore have been adopted in this study.

i

= ci +

n

(2)

tan ( i )

where i , ci , and i are shear strength, cohesion and friction angle of intact rock, respectively. Similarly, the shear strength of rock mass can be expressed as: rm

= crm +

n

tan (

(3)

rm )

where rm , crm , and rm are shear strength, cohesion and effective friction angle of rock mass, respectively. In rock engineering, failure criteria are typically expressed in terms of principal stresses. The Mohr-Coulomb failure criterion can be given by: 1

= 2c tan

2

+

4

+

3

tan2

2

+

4

(4)

where 1 and 3 are the major and minor effective principal stresses at failure, respectively while c and take appropriate values depending on the application (e.g., for intact rock c = ci , = i and for joints c = cj , = j ). 2.2. Hoek-Brown failure criterion The linear Mohr-Coulomb failure criterion provides a reasonable fit to experimental data when the range of normal stress during shear testing of natural joints or confining stress in triaxial testing of intact rock is relatively narrow. However, experimental failure envelopes of rocks are generally nonlinear over a wide range of stresses (e.g.,

Fig. 1. Effect of scale on transition from intact rock to heavily jointed rock mass, modified after Hoek and Brown (1980). 2

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Bieniawski, 1974; Hoek and Brown 1980). The empirical Hoek-Brown criterion is the most commonly used nonlinear failure criterion for rocks. Hoek and Brown (1980) showed that the failure envelope of intact rocks can be described using: 1, i

=

3

+

mi

c

3 c

with

= tan

0.5

=

3

+

mj

c

3 c

(5)

0.5

+ sj

=

3

+

c

mrm

3 c

+ srm

(6)

arm

(7)

where 1, rm is rock mass strength, mrm , srm and arm are rock mass constants controlling the slope, intercept and curvature of the failure envelope, respectively. It can be observed that the Hoek-Brown failure criterion is typically expressed in terms of effective principal stresses. Hoek (1983) reported the exact equations derived by Dr. John Bray of Imperial College for expressing the Hoek-Brown failure criterion for intact rock and joints in terms of effective normal and shear stresses:

= [cot ( )

cos ( )]

m c 8

+

16(m n + s c ) cos 2 3m2 c

1 sin 3

1

1+

16(m n + s c ) 3m2 c

0.5

1.5

1

(9)

where may be interpreted as the instantaneous friction angle and m and s take appropriate values depending on the application (e.g., for intact rock m = mi , s = si and for joints m = mj , s = sj ). While the nonlinear form of the failure criterion proposed by Hoek and Brown (1980) and Hoek et al. (2002) is useful, a far more significant contribution was made by providing relationships for downgrading the strength of intact rock based on rock mass quality to estimate rock mass strength (Marinos et al., 2005; Rafiei Renani et al., 2019). Initially, the rock mass rating (RMR) classification proposed by Bieniawski (1976) and the rock mass quality index (Q) classification proposed by Barton et al (1974) were used for quantifying rock mass quality. The geological strength index (GSI) system was later developed by Hoek (1994), Hoek et al. (1995) and Hoek and Marinos (2000) as the primary method for quantifying rock mass quality for the Hoek-Brown failure criterion. Fig. 2a shows the GSI chart for jointed rock masses which is based on qualitative geological observations regarding the degree of fracturing and surface condition of joints. Despite the attempts made by Sonmez and Ulusay (1999), Cai et al. (2004), Russo (2009), Hoek et al. (2013) and others to quantify the GSI chart based on limited empirical evidence, the qualitative chart given by Hoek and Marinos (2000) is frequently used in practice. The latest equations for obtaining rock mass constants from intact rock properties and rock mass quality are (Hoek et al. 2002; Hoek and Brown, 2019):

where 1, j is joint strength, mj and sj are joint constants controlling the slope and intercept of the failure envelope, respectively. For typical joints with negligible cohesion, sj = 0 is used. Utilizing the results of tests on large scale samples of rock with varying degrees of disturbance and weathering, the failure criterion was applied to rock masses (Hoek and Brown, 1980, 2019; Hoek et al. 2002): 1, rm

4 1+

6

+ si

where 1, i is intact rock strength, c is the uniaxial compressive strength of intact rock, mi is a material constant controlling the slope of failure envelope, and si = 1 for intact rock. Hoek and Brown (1980) and Hoek (1983) also used the same form of equation to describe the strength of joints: 1, j

1

mrm = mi exp srm = exp

(8)

GSI 100 28 14D

GSI 100 9 3D

(10) (11)

Fig. 2. (a) Geological strength index (GSI) chart for jointed rock masses, modified after Hoek and Marinos (2000) and (b) effect of GSI on the failure envelope of undisturbed rock mass with σc = 100 MPa and mi = 20. 3

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arm =

1 1 + exp 2 6

GSI 15

exp

20 3

isotropy is approached. It implies that the strength of heavily jointed rock masses with near isotropic behavior must be independent of intact rock strength. This is, however, in contrary to well-established rock mass failure criteria such as the Hoek-Brown failure criterion in which intact rock strength properties directly affect rock mass strength. This is because real joints have finite lengths and rock mass failure involves not only joint slip but also intact rock failure. Hence, it is essential for any realistic model of rock mass to take into account the non-persistent nature of joints in rock masses.

(12)

where GSI is the geological strength index ranging from zero to 100 depending on rock mass quality and D is the disturbance factor ranging from zero to unity depending on the extent of disturbance induced by blasting and stress relaxation near the excavation boundary. The effect of excavation and blasting damage is beyond the scope of this study and therefore a D factor of zero has been adopted in the following to focus on the strength of undisturbed rock masses. To illustrate the effect of GSI on rock mass strength, failure envelopes obtained using the strength properties c = 100 MPa and mi = 20 are shown in Fig. 2b. It can be observed that decreasing GSI causes a reduction in the intercept and slope of rock mass failure envelope.

4. Rock mass model with non-persistent joints 4.1. Conceptual model Establishing the key role that intact rock bridges and blocks can play in the behavior of rock masses, an analytical model of rock masses with non-persistent joints is presented in this study. Let the grey element in Fig. 5 represent an isotropic sample of intact rock with unit side length and shear strength of i . Similarly, the red element represents a sample of rock with unit side length containing a central persistent horizontal joint. The shear strength of the jointed sample in horizontal direction is equal to the joint shear strength j . Now consider Ni grey intact samples and Nj red horizontally jointed samples put together in a random horizontal arrangement. This assembly is conceptually similar to a layer of rock mass containing nonpersistent joints because joints are terminated at the boundary of intact rock and jointed samples. Utilizing the equations of force equilibrium for individual samples in the assembly, it can be shown that the shear strength of the rock mass layer in horizontal direction can be given by:

3. Rock mass model with persistent joints Rock mass is a system of intact rock intersected by a network of joints. Therefore, an elementary model of rock mass can be obtained by considering a sample of rock containing a number of joints. Jaeger (1960) analyzed a sample of rock in triaxial compression containing a single persistent joint oriented at an angle with respect to the direction of major principal stress (Fig. 3a). Depending on the joint inclination, the sample can fail either due to intact rock failure or joint slip. The potential for intact rock failure can be evaluated using intact rock failure criteria and the possibility of joint slip can be assessed by comparing the stress and strength on the joint plane. Normal and shear stresses on the joint can be readily determined using stress transformation equations. Using a linear Mohr-Coulomb failure criterion for the joint, the major effective principal stress required to cause slip on the persistent joint is: 1

=

3

+

2[cj + [1

3

rm

tan ( j )]

tan ( j ) tan ( )] sin (2 )

(13)

=

Nj Ni i + Ni + Nj Ni + Nj

j

(14)

Joint persistence, p can be defined as the sum of individual joint surface areas to the total surface area of a coplanar reference plane (Dershowitz and Einstein, 1988; Cai et al., 2004):

Differentiation of Eq. (13) with respect to joint orientation reveals that the axial stress required for the joint to slip is minimum when equals /4 j /2 . The axial stress required for joint slip tends to infinity as approaches zero and /2 j where sample strength is controlled by the strength of intact rock. The variation of sample strength with joint orientation according to this model is shown in Fig. 3b. It can be observed that the strength of a sample with a single persistent joint is highly anisotropic varying from intact rock strength to joint strength. As shown in Fig. 1, a large scale representative sample of rock mass contains many joints at various orientations. The persistent joint model can be extended to analyze a sample with multiple joints by superimposing the joint slip curves and finding the overall minimum strength at each direction. As an example, Fig. 4 uses the results of triaxial tests carried out by McLamore and Gray (1967) on samples of slate containing parallel planes of weakness. It can be observed that the single persistent joint model captures the variation of strength reasonably well. Using the same strength properties for intact rock and weak plane, variation of strength with two, three, and four joints at equal angular distances is obtained. Fig. 4 shows that as the number of joints increases, the overall strength of the sample becomes less and less anisotropic because joint slip occurs over a wider range of directions. Note that in the sample with three persistent joints at 60° angles, failure is always due to slip on the joints. With four joints at 45° angles, the variation of strength with direction is small enough to justify the assumption of isotropy (Hoek and Brown, 1980; Brady and Brown 2006). The reduction of strength anisotropy with increasing the number of joints is the basis for using isotropic failure criteria for jointed rock masses at large scale. However, it should be noted that according to the persistent joint model, the overall sample behavior is fully controlled by joint slip and joint strength properties well before the condition of

p=

Nj Jointed area = Total area Ni + Nj

(15)

Therefore, Eq. (14) can be re-written as: rm

= (1

p) i + p

j

(16)

Eq. (16) implies that rock mass strength is a weighted average of the strength of its elements with (1 p) and p as the weight coefficients of intact rock strength and joint strength, respectively. It satisfies the limiting conditions of rm = i when joint persistence is zero (no joints)

Fig. 3. (a) Configuration of a sample with persistent joint in triaxial compression and (b) variation of strength by joint angle at constant confining stress. 4

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Fig. 4. Results of triaxial tests on slate at effective confining stress of 35 MPa (after McLamore and Gray, 1967) and variation of strength based on the persistent joint model with (a) one, (b) two, (c) three and (d) four joints at equal angular distances. rm

= Mi (1

p) i + Mj p

j

(17)

where Mi and Mj are intact rock and joint strength mobilization factors, respectively. The strength mobilization factors are dimensionless and range from zero to unity depending on the extent to which the strength of intact rock bridges and joints are mobilized to resist the shear failure of rock mass. The strength mobilization factors are influenced by deformation characteristics of intact rock and joints which in turn depend on factors such as stress condition. As a result of such complex interactions, direct determination of the strength mobilization factors using analytical approaches is cumbersome and beyond the scope of this study. Full mobilization of intact rock strength and joint strength can be represented by Mi = Mj = 1 which reduces Eq. (17) to Eq. (16). This is the assumption used in the remainder of this paper. Note that joint persistence defined in Eq. (15) is based on the surface area of joints which is difficult to measure in the field. Joints are typically observed as linear features on exposed surfaces of rock. Therefore, it is much easier to calculate a length-based persistence pl as the sum of joint trace lengths divided by the total length of a colinear scan line (Dershowitz and Einstein, 1988). Using geometric relationships between lengths and areas, it can be shown that the area-based joint persistence p used in the analytical model can be approximated by pl2 . Eqs. (16) and (17) provide a versatile framework for developing rock mass failure criteria using intact rock and joint failure criteria.

Fig. 5. Model of rock mass with non-persistent joints as an assembly of randomly arranged intact and jointed elements.

and rm = j with joint persistence of unity (persistent joint). Eq. (16) is based on the simplifying assumption that the strength of intact rock and joints are fully and simultaneously mobilized to resist the excessive shear deformation and failure of the rock mass. In that sense, it provides an upper bound for rock mass strength. In reality, the process of rock mass failure is more complex and full mobilization of intact rock strength and joint strength may not occur simultaneously owing to different deformation characteristics of intact rock and joints. Therefore, the general equation for the strength of rock mass based on the non-persistent joint model can be given by:

Fig. 6. Variation of strength based on the non-persistent joint model with (a) one, (b) two, (c) three and (d) four joints at equal angular distances. 5

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Such rock mass failure criteria may be categorized as associated failure criteria in which the same type of failure criterion is used for both intact rock and joints and non-associated failure criteria obtained using different types of failure criteria for intact rock and joints. The terms associated and non-associated failure criteria coined in this study are not to be confused with the associated and non-associated flow rules used in the theory of plasticity for determination of plastic strains. In the following, linear and nonlinear forms of associated rock mass failure criteria are developed using the Mohr-Coulomb and Hoek-Brown failure criteria, respectively. Following the same methodology, other associated and non-associated rock mass failure criteria can be developed.

It can be observed from Fig. 6 that as joint persistence increases, the minimum strength of the jointed sample decreases and strength anisotropy generally increases. At any given level of joint persistence, strength anisotropy decreases by increasing the number of planes of weakness at equal angular distances. The sample with four planes of non-persistent joints at 45° angles is approaching the condition of isotropy with relatively small variation of strength with direction. For a large scale representative sample of rock mass containing many randomly oriented non-persistent joints with comparable sizes and strength properties, it is reasonable to assume an isotropic behavior. The behavior of such a sample is governed by rock mass shear strength parallel to the plane of non-persistent joints as given by Eqs. (16)–(19). Therefore, rock mass strength based on the linear non-persistent joint model can be expressed in terms of effective principal stresses as:

4.2. Linear analytical failure criterion The linear equations of the non-persistent joint model can be obtained by using the Mohr-Coulomb failure criterion for intact rock and joints. Substituting Eqs. (1)–(3) into Eq. (16) gives the equations for rock mass strength properties:

crm = (1 rm

= tan

p) ci + pcj 1

[(1

p) tan ( i ) + p tan ( j )]

1, rm

=

3

+

[1

2[crm + 3 tan ( rm )] tan ( rm ) tan ( )] sin (2 )

rm

2

+

4

+

3

tan2

rm

2

+

(21)

4

(18)

where crm and rm are obtained from Eqs. (18) and (19) based on the non-persistent joint model. Eq. (21) is similar to Eq. (4) when used for rock mass and is given here for the sake of completeness.

(19)

4.3. Nonlinear analytical failure criterion

Note that Eqs. (18) and (19) describe the strength properties of rock mass in shear parallel to the plane of non-persistent joints. In a general triaxial loading condition, the condition of slip along the plane of nonpersistent joints can be given by: 1

= 2crm tan

The linear expression of the Mohr-Coulomb failure criterion in terms of normal and shear stresses can be conveniently incorporated in the equations of the non-persistent joint model to determine rock mass strength properties using Eqs. (18) and (19). With the complex form of the nonlinear Hoek-Brown failure criterion in terms of normal and shear stresses given in Eqs. (8) and (9), it is not possible to develop simple explicit equations for rock mass strength properties. The nonlinear form of the non-persistent joint model can be obtained by substituting Eq. (8) in Eq. (16):

(20)

Eq. (20) is obtained by satisfying the shear failure criterion on the plane of non-persistent joints oriented at an angle with respect to the direction of the major principal stress. This is similar to Eq. (13) except that joints are non-persistent and therefore the shear strength on the plane of weakness depends not only on joint strength but also on intact rock strength and joint persistence. Using the example shown in Fig. 4, the effect of joint persistence on the strength of jointed samples is shown in Fig. 6. For the case of zero joint persistence, sample strength is isotropic and equal to intact rock strength. Using the joint persistence of unity, the results of the nonpersistent joint model coincide with those of the persistent joint model given by Jaeger (1960). Hence, the non-persistent joint model satisfies the analytical solutions for the strength of fracture-free and fully jointed rocks and provides a reasonable transition between the two extremes.

rm

=

c

8

[mi (1

p)(cot ( i )

cos ( i )) + mj p (cot ( j )

cos ( j ))]

(22)

where i and j can be determined by using (mi , si ) and (mj , sj ) in Eq. (9), respectively. Eq. (22) expresses rock mass strength in terms of normal and shear stresses at failure. In rock engineering applications, failure criteria are typically expressed in terms of principal stresses. It is cumbersome to analytically derive such explicit expression from Eq. (22) and therefore a numerical approach is adopted. Eq. (22) can be used to generate closely spaced ( n, rm ) data points on the rock mass failure envelope. Using the analytical solution for Mohr’s envelope derived by Balmer (1952),

Fig. 7. Effect of joint persistence on rock mass failure envelope using (a) the linear analytical failure criterion with φj = 30° and (b) the nonlinear analytical failure criterion with mj = 0.3. 6

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corresponding major and minor effective principal stresses can be given by: 1

=

n

3

=

n

+

rm

rm

1

cos (2 ) sin (2 )

(23)

1 + cos (2 ) sin (2 )

(24)

with

=

1 tan 2

1

d rm + d n 4

(25)

where may be interpreted as the angle between the plane of failure and the direction of minor effective principal stress. Considering the complex functional relationship for rm given in Eq. (22), the analytical approach for solving the differentiation d rm in Eq. (25) is cumbersome. d n

However, it is possible to utilize the closely spaced ( n, rm ) data points in numerical differentiation to approximate d rm with rm . Consistent d n n with the findings of Carranza-Torres (2004), this method of approximation was found to give sufficiently accurate results.

Fig. 9. Ratio of uniaxial compressive strength of rock mass to intact rock in terms of joint persistence and GSI.

by the empirical Hoek-Brown failure criterion (Fig. 2b). Since the empirical Hoek-Brown approach of estimating rock mass strength using the GSI system is widely used in rock engineering, it is useful to compare the results of the proposed analytical model with those of the empirical approach. An important factor to consider is the ratio of rock mass strength to intact rock strength in uniaxial compression which indicates the degree by which strength is degraded due to the presence of joints. Fig. 9 compares the strength ratios obtained from the empirical Hoek-Brown criterion and the linear analytical failure criterion. While the relationship between the strength ratio and GSI in the Hoek-Brown failure criterion is highly nonlinear, joint persistence in the analytical model has an almost linear relationship with the strength ratio which is rather insensitive to joint friction angle. It can be observed that joint persistence used in the analytical model has not only a physical definition as the normalized jointed area, but also a mechanical interpretation as the normalized reduction in uniaxial compressive strength due to the presence of joints. For instance, a joint persistence of 0.8 implies an approximate 80% reduction in uniaxial compressive strength compared with intact rock strength and therefore a strength ratio of 0.2. This interpretation becomes more accurate as joint persistence approaches zero or unity and joint friction angle increases. To compare the empirical and analytical failure criteria in confined

5. Results 5.1. Rock mass strength The linear and nonlinear equations of the non-persistent joint model make it possible to address some fundamental questions about the impact of joints on the strength of rock masses. More specifically, the model provides an analytical framework to quantify the extent by which intact rock strength should be degraded depending on the joint persistence and shear strength to arrive at rock mass strength. In the following, an example is presented to illustrate the application of the analytical failure criteria. Intact rock properties of ci = 18.2 MPa, i = 50°, c = 100 MPa and mi = 20 were used. Because the cohesive strength is typically much smaller than the frictional strength for rock joints, cj =sj = 0 was adopted. Joint persistence and friction angle were varied over a wide range to investigate the degradation of rock mass strength. Figs. 7 and 8 show the effect of joint persistence and shear strength on rock mass failure envelope using the linear and nonlinear analytical failure criteria. It can be observed that increasing joint persistence and decreasing joint strength cause a reduction in the cohesive and frictional components of rock mass strength. This is evident from the reduction in the intercept and slope of rock mass failure envelopes. This is a reasonable trend also supported

Fig. 8. Effect of joint strength on rock mass failure envelope using (a) the linear analytical failure criterion, and (b) the nonlinear analytical failure criterion with joint persistence p = 0.75. 7

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conditions, the Hoek-Brown failure envelopes were developed using the same intact rock strength properties and fitted to the analytical failure envelopes by adjusting the value of rock mass GSI. As an example, Fig. 10 shows the linear and nonlinear analytical failure envelopes for a joint persistence of 0.5 and the equivalent empirical Hoek-Brown failure envelope obtained using the GSI value of 78. It can be observed that the empirical failure envelope is in reasonable agreement with the analytical failure envelopes especially with the linear failure envelope. The empirical failure criterion tends to give relatively lower strength values at the two ends of the confining stress range and higher values for intermediate levels of confinement.

However, most rock joints in the field have intermediate values of friction angle. This suggests that the range of rock mass quality typically encountered in the field corresponds to a window inside the analytical chart with more intermediate levels of fracture intensity and joint strength properties. This is shown as the shaded area in Fig. 11. The analytical GSI chart of Fig. 11 is of fundamental importance for two major reasons. First, it provides the very first validation of the commonly used empirical GSI chart over the full range of rock mass conditions based on analytical solutions rather than limited empirical evidence. Second, it eliminates one of the major shortcomings of the previous empirical GSI charts identified by Hoek et al. (2013) by using shear strength properties of joins rather than qualitative descriptions or arbitrary ratings of joint surface conditions. The analytical GSI chart is also of practical value as it facilitates the determination of equivalent joint persistence in the analytical failure criteria for rock masses characterized by the GSI system. This is illustrated using a practical example in Section 7.

5.2. Analytical GSI chart The empirical GSI chart in Fig. 2a uses the degree of fracturing on the vertical axis and joint surface condition on the horizontal axis. The value of GSI in turn governs the degree to which intact rock strength is downgraded due to the presence of such fractures to arrive at rock mass strength. The degree of fracturing on the vertical axis of the empirical GSI chart is conceptually similar to joint persistence used in the analytical model. Joint persistence is also a measure of block size, used by Cai et al. (2004) and Russo (2009) for quantification of the GSI chart. Joint surface condition on the horizontal axis of the empirical GSI chart governs joint shear strength which is also present in the analytical model. It is therefore possible, in principle, to develop a similar GSI chart based on the analytical non-persistent joint model. Developing such a quantitative GSI chart, in practice, requires finding the contours of GSI in terms of joint persistence and joint friction angle. Following the approach shown in Fig. 10 using a wide range of joint persistence and shear strength properties gave a series of data points containing the equivalent GSI for various joint parameters in the analytical model. Interpolating the data points allowed the construction of the quantitative GSI chart calibrated to the results of the analytical model (Fig. 11). Despite the drastically different approaches used in the development of the empirical GSI chart in Fig. 2a and the analytical GSI chart in Fig. 11, there is generally a reasonable agreement between the GSI contours in these charts. The empirical GSI contours are linear and almost equally spaced whereas the analytical GSI chart contains nonlinear contours at varying distances. This characteristic of the analytical GSI chart is consistent with other quantitative GSI charts proposed by Sonmez and Ulusay (1999), Cai et al. (2004) and Russo (2009) which adopt nonlinear scales on the axes. However, no new insight can be gained from comparing the absolute location of GSI contours in the analytical GSI chart and the empirical GSI chart because the latter is based on qualitative descriptions and therefore the trend and distance between the contours do not correspond to any specific changes in physical quantities. It is useful for practical applications to establish relationships between the qualitative descriptions adopted in the empirical GSI chart and the quantitative parameters used in the analytical model. Descriptions of joint surface condition adopted in the empirical GSI chart can be matched with those used in the ISRM (1981) classification of joint friction angles. This is shown in Table 1 which relates the horizontal axes on the empirical and analytical GSI charts. Comparing the empirical and analytical GSI charts, it is also possible to establish approximate relationships between the descriptions of rock mass structure and joint persistence. Table 2 relates the vertical axes on the empirical and analytical GSI charts and can be used to classify joint persistence based on the descriptions of rock mass structure. Note that the analytical GSI chart covers joint persistence values ranging from zero to unity corresponding to fracture-free and completely fractured rock masses, respectively. In reality, most rock masses have an intermediate degree of fracturing. Joint friction angle in the analytical GSI chart ranges from 10 to 50° which according to the ISRM (1981) correspond to “very low” and “very high” values, respectively.

6. Validation In order to validate a failure criterion, a series of high quality standard tests have to be carried out on several carefully prepared undisturbed samples under various loading conditions. This is a relatively straightforward task when dealing with manufactured engineering materials and even intact soil and rock at small scale. For jointed rock masses, however, obtaining a sufficient number of large scale undisturbed samples and carrying out standard tests at various loading conditions are prohibitively challenging. As a result, reliable strength measurements of naturally jointed rock masses at large scale are extremely rare. This indeed has been a major obstacle in gaining a better insight into the behavior of jointed rock masses and making tangible progress in estimating rock mass strength. The difficulties associated with testing of representative samples of naturally jointed rock masses have inspired studies on physical models typically comprised of manufactured materials with artificial joints of various simplified configurations (e.g., Einstein et al., 1969; Brown, 1970; Brown and Trollope, 1970; Einstein and Hirschfeld, 1973; Kulatilake et al., 1997). However, mechanical characteristics of the model material as well as geometric configuration of the artificial joint networks are often significantly different from those of actual intact rock and natural joints. Therefore, the relevance of results obtained from such manufactured models to rock masses with complex natural joint networks is debatable. The most comprehensive and reliable set of published triaxial test results on naturally jointed rock masses known to the authors are those

Fig. 10. Analytical rock mass failure envelopes (I and II denote linear and nonlinear forms) with p = 0.5, φj = 40°, mj = 2, and the equivalent empirical Hoek-Brown failure envelope with GSI = 78. 8

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Fig. 11. Quantitative GSI chart based on the analytical model of rock mass with non-persistent joints. Table 1 Relationship between joint surface condition and joint friction angle. Surface condition in the GSI chart

ISRM description of joint friction angle

Joint friction angle (°)

Very good Good Fair Poor Very poor

Very high High Moderate Low Very low

> 45 35–45 25–35 15–25 < 15

Table 2 Relationship between rock mass structure and joint persistence.

reported by Jaeger (1970). This is the same set of data upon which the Hoek-Brown rock mass failure criterion and relationships between intact rock and rock mass strength in terms of rock mass quality have

Structure in the GSI chart

Joint persistence

Intact/Massive Blocky Very blocky Blocky/Disturbed/Seamy Disintegrated Laminated/Sheared

< 0.60 0.60–0.75 0.75–0.80 0.80–0.85 0.85–0.90 > 0.90

been founded. The tests were carried out on Panguna andesite from the island of Bougainville in Papua New Guinea. As described by Hoek and Brown (1980), this material was the host rock for a major copper 9

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deposit and the large scale of the mining operation afforded an unusually large amount of testing to be carried out. The results of tests on Panguna andesite were used to obtain the strength properties of intact rock and joints given in Table 3 (Jaeger, 1970; Hoek and Brown, 1980). Fig. 12a shows the results of triaxial compression tests on undisturbed samples of jointed Panguna andesite. It can be observed that the rock mass strength is significantly lower that the intact rock strength due to the high degree of fracturing in this rock mass. Analytical rock mass failure envelopes were obtained using a joint persistence value of 0.94. This value of joint persistence is consistent with the heavily jointed nature of the samples of Panguna andesite and provides reasonably accurate rock mass failure envelopes. The empirical failure envelope obtained by Hoek and Brown (1980) for the undisturbed samples of jointed Panguna andesite is also shown in Fig. 12a. It can be observed that beyond the confining stress of 10 MPa, the empirical criterion begins to underestimate the strength of the samples. This is because Hoek and Brown (1980) fitted the failure envelope to data with low confining stresses. This is by no means a shortcoming of the Hoek-Brown failure criterion as it is possible to obtain a better overall fit using data with a wider range of confining stress. Another set of comprehensive and reliable data is obtained using the pioneering experimental approach introduced by Rosengren and Jaeger (1968), and utilized by Gerogiannopoulos (1977). In these experiments, intact samples of marble were subjected to extreme temperatures causing the formation of microcracks in the samples due to nonuniform expansion of different grains. This method provides undisturbed samples of rock with thermally induced microcracks whose configuration is governed by the granular structure of the material. As a result, samples obtained from this method are perhaps the most realistic physical models of tightly interlocked rock masses. The results of tests on Carrara marble gave the strength properties of intact rock and joints given in Table 3 (Gerogiannopoulos, 1977; Hoek, 1983). Fig. 12b shows the results of triaxial compression tests on thermally granulated samples of Carrara marble. It can be observed that the strength of granulated samples is slightly lower than intact rock strength due to the extremely high degree of granular interlocking. Analytical rock mass failure envelopes were obtained using a joint persistence value of 0.15. This value of joint persistence is consistent with the tightly interlocked structure of the granulated Carrara marble and generally provides reasonable estimates of sample strength. Note that the empirical failure envelope obtained by Hoek (1983) for the granulated samples of Carrara marble coincides almost exactly with the nonlinear analytical failure envelope (Fig. 12b). The strength of the granulated samples tested at confining stresses of below 2 MPa is considerably overestimated by the failure envelope passing through other data points. It is likely that at very low levels of confining stress, failure is mainly governed by the strength of cracks and joints and the strength of intact rock is not fully mobilized until higher levels of confinement are reached. This phenomenon can be accommodated by the general model of rock mass with non-persistent joints using a stress-dependent intact rock strength mobilization factor Mi in Eq. (17) with a near-zero value at unconfined condition which gradually increases with confining stress and approaches unity at confinement levels of above 2 MPa. A detailed discussion of the strength

mobilization factors at very low confinements requires further experimental and analytical studies which are beyond the scope of this study. 7. Practical application To demonstrate the application of the proposed model, the response of a large cavern in a jointed rock mass is analyzed using the linear analytical failure criterion. The powerhouse cavern in the Nathpa Jhakri Hydroelectric project is located 500 m from the Sutlej River in Himachal Pradesh, India. The cavern is 20 m wide, 49 m high and 216 m long with about 262 m of overburden at the crown. The opening was excavated in 12 stages and multipoint borehole extensometers (MPBX) were installed at various sections along the cavern axis and locations around the excavation periphery to monitor the performance of the cavern (Fig. 13). The rock mass around the cavern mainly consists of jointed quartz mica schist. The in situ condition and properties of the rock mass are extensively studied and the performance of the cavern is analyzed using various continuum and discontinuum models (e.g., Bhasin et al., 1995, 1996; Varadarajan et al., 2001; Sitharam and Madhavi Latha, 2002; Al-Obaydi et al., 2008). The results of in situ stress measurements using the hydraulic fracturing method indicated a gravitational stress field with a ratio of in-plane horizontal to vertical stress of 0.8 (Bhasin et al., 1996; Varadarajan et al., 2001). Rock mass deformation modulus of 6.7 GPa and Poisson’s ratio of 0.2 were reported by Varadarajan et al. (2001). Hoek and Brown (1997) reported intact rock strength properties including c = 30 MPa and mi = 15.6 which are equivalent to ci = 4.80 MPa and i = 51.8° for an underground excavation at this depth (Hoek et al., 2002). They also assigned a GSI value of 65 to the rock mass around the cavern. Considering the joint properties including the residual friction angle of 25° and a joint roughness coefficient of 5 reported by Bhasin et al. (1996), it is reasonable to use j = 30°. Using the values of GSI and j in the analytical GSI chart of Fig. 11, the equivalent joint persistence of 0.61 is obtained for this rock mass. According to the analytical model, rock mass strength properties can be calculated as crm = 1.87 MPa and rm = 40.4° using Eqs. (18) and (19). The response of the powerhouse cavern was analyzed using the finite element analysis software, RS2 (Rocscience Inc., 2018). Due to the symmetric configuration of the problem, only half the cavern was modelled for numerical analysis (Varadarajan et al., 2001; Sitharam and Madhavi Latha, 2002; Al-Obaydi et al., 2008). A fine resolution mesh with quadratic quadrilateral elements was used to discretize the model. Numerical analysis was carried out in 12 stages reflecting the actual sequence of excavation as shown in Fig. 13. Fig. 14 shows the distribution of the major effective principal stress on deformed meshes at various stages of analysis. The low stress region formed at the cavern wall coincides with the plastic zone. Maximum horizontal displacement at the wall is 44 mm and maximum vertical displacement at the crown is 17 mm. These values are in close agreement with those obtained from other numerical studies (Bhasin et al., 1996; Varadarajan et al., 2001; Al-Obaydi et al., 2008). Displacements measured by the extensometers can be compared with those obtained from the model for further validation. As pointed out by Rafiei Renani et al. (2016), it is crucial to apply temporal and spatial corrections to total model displacements before comparing with

Table 3 Strength properties of Panguna andesite and Carrara marble used in the analytical model. Intact rock strength properties Mohr-Coulomb Rock

ci (MPa)

Panguna andesite Carrara marble

46.0 21.5

Joint strength properties Hoek-Brown

i (°)

52 36

Mohr-Coulomb

c (MPa)

265 82

10

mi

cj (MPa)

18.9 8.7

1.7 0

Hoek-Brown j (°)

29.7 28.0

mj

sj

0.5 2.0

0.0002 0

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Fig. 12. Validation of the analytical rock mass failure criteria (I and II denote linear and nonlinear forms) using the results of tests on (a) naturally jointed Panguna andesite, after Jaeger (1970) and (b) thermally granulated Carrara marble, after Gerogiannopoulos (1977).

that the overall response of the rock mass to induced stresses around the cavern is adequately captured by the analytical model for rock mass strength. Analysis of the Nathpa Jhakri powerhouse cavern is presented here as a simple example of the application of the analytical model rather than an exhaustive case study of the site specific conditions at that project. Future studies are required utilizing the results of additional laboratory experiments and in situ measurements to further validate the proposed analytical model and make it a viable tool in practical rock engineering. 8. Conclusions The effect of fractures on rock mass strength was discussed and the importance of considering the non-persistent nature of joints was highlighted. An analytical model of rock mass with non-persistent joints was presented and used to develop linear and nonlinear failure criteria for rock masses in terms of intact rock strength, joint shear strength and joint persistence. Analyzing the anisotropic behavior of samples with few joints showed that the analytical model matches the solutions for the strength of fracture-free and fully fractured rocks and provides a reasonable transition of strength for rocks with non-persistent joints. The analytical model made it possible to systematically study the effect of joint intensity and joint shear strength on the strength of rock masses at large scale. The model showed that increasing joint persistence and decreasing joint shear strength cause a reduction in the cohesive and frictional components of rock mass strength. Considering the ratio of rock mass strength to intact rock strength, it was shown that joint persistence used in the analytical model has not only a physical definition as the normalized jointed area but also a mechanical interpretation as the normalized reduction in strength due to the presence of joints. The relationship between the analytical model and the empirical Hoek-Brown approach for estimating rock mass strength was explored. It was shown that compared with the analytical model, the Hoek-Brown failure criterion generally gives higher strength at intermediate levels of confining stress and lower strength at two ends of confinement range. Using the analytical failure criteria, a quantitative GSI chart in terms of joint persistence and joint shear strength was developed and calibrated. The trend of contours in the analytical GSI chart is reasonably similar to that in the empirical chart. The results of comprehensive tests on undisturbed samples of

Fig. 13. The sequence of excavation and layout of extensometers at the Nathpa Jhakri powerhouse cavern, modified after Varadarajan et al. (2001).

extensometer measurements. The temporal correction is applied to take into account the timing of extensometer installation. The spatial correction is applied to obtain the relative displacement between various anchor points along the extensometer. Table 4 shows the range of displacements measured by various extensometers and those obtained from the model. It can be observed that the predicted displacements are within the range of measured displacements in all cases except for extensometer A. Contrary to model results, the measurements indicate a larger displacement in extensometer A compared with extensometer B due to widening of the central drift in the stage 2 of excavation. Considering the location of extensometers A and B relative to the area under excavation in the stage 2 (Fig. 13), the modeling results appears reasonable predicting larger displacements for extensometer B which is right below the area under excavation compared with extensometer A which is farther away. The larger displacements recorded by extensometer A are likely due to gravity-driven local movement of blocks of rock formed at the crown of the cavern. Capturing the local gravity-driven movement of individual blocks of rock is beyond the scope and capabilities of continuum models which are concerned with the overall behavior of rock mass in response to stresses. The results of numerical analysis indicate 11

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Fig. 14. Distribution of major principal stress around the Nathpa Jhakri powerhouse cavern.

combined with additional experimental results, can lead to a better insight into the behavior of jointed rock masses.

Table 4 Measured and predicted displacements for the extensometers at the Nathpa Jhakri powerhouse cavern. MPBX

During excavation stage(s)

Measured displacement (mm)

Predicted displacement (mm)

A B C D E

2 2 6 7–12 11–12

13–18 6–12 1–4 10–45 1–3

3.4 8.2 1.2 13.9 1.6

Acknowledgements This work was financially supported by the Natural Sciences and Engineering Research Council of Canada (NSERC: RES 0014117, CRD 470490-14), Canadian Nuclear Waste Management Organization (NWMO), and Rio Tinto. Declaration of Competing Interest

naturally jointed Panguna andesite and thermally granulated Carrara marble were used to validate the analytical model. The analytical failure envelope of Panguna andesite was obtained using a relatively high value of joint persistence consistent with the heavily jointed nature of the rock mass. On the other end of the spectrum, a relatively low value of joint persistence was used to obtain the failure envelope of the tightly interlocked granulated Carrara marble. Practical application of the analytical rock mass model was illustrated using numerical analysis of the Nathpa Jhakri powerhouse cavern. The results were found to be in reasonable agreements with field measurements and those obtained in other studies. It is obvious that the geometric configuration of natural joint networks as well as the mechanical interaction between joints and intact rock bridges is far more complex than those assumed in the analytical model. Nonetheless, the analytical model presented in this study proved to be an effective tool in studying the influence of joint intensity and joint shear strength on the strength of jointed rock masses. The presented linear and nonlinear failure criteria and the analytical GSI chart are by no means suggested to replace the existing empirical methods of estimating rock mass strength but rather to supplement and support the existing approaches. The methodology presented in this study can serve to develop more sophisticated and realistic models of rock mass, which

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